On probability of high extremes for product of two independent Gaussian stationary processes

Abstract

Let X(t), Y(t), t≥0, be two independent zero-mean stationary Gaussian processes, whose covariance functions are such that r _i (t)=1−|t|^ai+o(|t|^ai) as $t\rightarrow 0$, with 0<a _i ≤2, i=1,2 and both of the functions are less than one for non-zero t. We derive for any p the exact asymptotic behavior of the probability $P(\max _{t\in \lbrack 0,p]}X(t)Y(t)>u)$ as $u\rightarrow \infty$. We discuss possibilities generalizing obtained results to other Gaussian chaos processes h(X(t)), with a Gaussian vector process X(t) and a homogeneous function h of positive order.